Compute $\tan \left (\operatorname{arccot} \frac{4}{7} \right).$
Answer: Consider a right triangle where the adjacent side is 4 and the opposite side is 7.

[asy]
unitsize (0.5 cm);

draw((0,0)--(4,0)--(4,7)--cycle);

label("$4$", (2,0), S);
label("$7$", (4,7/2), E);
label("$\theta$", (0.8,0.5));
[/asy]

Then $\cot \theta = \frac{4}{7},$ so $\theta = \operatorname{arccot} \frac{4}{7}.$  Hence, $\tan \theta = \frac{1}{\cot \theta} = \boxed{\frac{7}{4}}.$